Theory of perturbation

These rules follow directly from the quantum-mechanical theory of perturbations and the resolution of the secular equations for the orbital interaction problem. The (small) interaction between orbitals of significantly different energ is the familiar second order type interaction, where the interaction energy is small relative to the difference between EA and EB. The (large) interaction between orbitals of same energy is the familiar first order type interaction between degenerate or nearly degenerate levels. 

The energy of the diatomic molecule, as given by Bq. (79) does not take into account foe anharmonicity of the vibration. Tte.qjE fect of fop cubic and qoartic terms in Eq. (73) can be evaluated by application of the theory of perturbation (see Chapter 12). 

Formally the model [41] is a specific case of applying the theory of perturbations of MOs to their special class of eq. (4.20). 

Dewar s theory. In recent years more and more popular in organic chemistry becomes the theory of perturbation molecular orbitals (PMO) worked out by the American chemist M.J.S. Dewar. Using that theory, many complex problems of organic chemistry can be tackled, including the problem of aromaticity. Since it is impossible to describe this theory here, we are citing only some of the conclusions concerning the cyclic polyenes, drawn from it. 

In the helium atom two electrons revolve about the nucleus (nuclear charge 2e) we have therefore 6 co-ordinates to deal with instead of 3, with the result that an exact solution is no longer possible. For the purpose of obtaining a general idea of the possible states, an exact solution is, however, not at all necessary following Bohr, we can in the first place neglect the mutual interaction of the electrons, and for a first approximation treat the problem as if the two electrons moved undisturbed in the field of the nucleus. Afterwards, the interaction can be taken into account by the methods of the theory of perturbations. 

From the point of view of macroscopic electrodynamics the appearance of a constant current under the influence of light (photovoltaic effect) can be understood in the framework of the theory of perturbation (for a small light intensity) from the nonlinear relation (see also (28)) ... 

An attempt has actually been made to subject to the theory of perturbations the second simplest atom, that of helium, with its one nucleus and two electrons. The result, however, was entirely negative the discrepancies between theory and observation were much too large to be accounted for by the inaccuracy of the calculations. This indicates that there is some basic error in the principles of our atomic mechanics. 

In order to set about this thoroughly, it is necessary to follow through the method suggested, and to examine the consequences to which we are led by the application of classical mechanics in conjunction with the quantum restrictions. We shall therefore give in this chapter a detailed account of the theory of perturbations, in-... 

We are of opinion that this will not be labour spent in vain, but that this broad development of the theory of perturbations will, together with the negative results, form the foundation for the true quantum theory of the interaction of several electrons.1... 

Problems of this kind are dealt with in celestial mechanics, and the various methods adopted for their solution are referred to under the heading Theory of Perturbations. The additional terms AH1+AaH2+. . . are in fact regarded as a perturbation of the unperturbed motion characterised by H0. 

Equations of the same type occur very frequently in the theory of perturbations. To integrate them, A and B are each separated into a constant part, depending only on the J s, and a purely periodic component ... 

We find, then, that a systematic application of the theory of perturbations does not lead to a satisfactory model of the normal helium atom. It might be supposed that the failure of our method was due to the fact that we are dealing here with the normal state, where several electrons move in equivalent orbits, and that a better result would be anticipated in the case of the excited states, where the main characteristics of the spectra are reproduced by the quantum theory in the form used here. We shall now show that this again is not the case. 

We shall now investigate the highly excited orbits of the helium atom on the basis of the theory of perturbations, by which we mean the external orbits which can be occupied by an electron when added to a helium ion. We shall assume that the orbit of the first electron in the ion is circular. Our problem is to investigate those types of orbits for which the inner electron, if unperturbed, would move in a one-quantum circle. 

After calculating the unperturbed motion of the inner electron, we can find the secular motions of the remaining variables by introducing a new Hamiltonian function, the mean value of Hx taken over the unperturbed motion of the inner electron. The integration of the corresponding Hamilton-Jacobi equation is again performed by the methods of the theory of perturbations. 

The partial differential equation H1=const. is not separable. Since, however, it may be resolved into terms of different orders of magnitude, it can be /lealt with by the methods of the theory of perturbations. Let us put... 

The BAC-MP4 method allows the enthalpy of formation of a chemical species to be estimated. The calculation of electronic structures (geometry and vibration frequencies) can be carried out using the GAUSSIAN 94 computer program. The corrections are carried out using the Mdller-Plasset (MP) theory of perturbations and the Bond Additivity Corrections are also carried out using the same theory. 

L. N. Ussachoff, Equation for the importance of neutrons, recwtor kinetics and the theory of perturbation. Proceedings of the International Conference on the Peaceful Uses of Atomic Energy, vol. 5, 1956. 

The effect of strains on the band structure of crystals has been intensively investigated by Bir and Pikus, using both the theory of perturbation and a group-theoretical method, namely the theory of invariants. The approach can be applied to deduce the strain derivatives of the wavefunctions appearing in Eq. (5). Following the procedure of the authors of Ref. 8, the Kohn-Sham operator for a valence electron in a strained crystal can be written in the form... 

In the Zimm theory of perturbations known as first-order perturbations, where only the interactions between segments are taken into consideration, the relation... 

---